Precalculus (6th Edition) Blitzer

a) The percentage model function for African Americans who used cigarettes is $y=38-0.54x$. b) In the year $1993$, the use of cigarettes for both groups was the same. The percentage of use of cigarettes for African Americans in 1993 is, $\text{33}\text{.68 }\!\!%\!\!\text{ }$. The percentage of use of cigarettes for Hispanics in 1993 is, $\text{33}\text{.68 }\!\!%\!\!\text{ }$.
(a) We use the model function of cigarette use by African Americans after 1985: $y=38-0.54x$ (b) Step 1: Consider the percentage model function for African Americans who used cigarettes after x years from $1985$: $y=38-0.54x$ (I) Step 2: The percentage model function for Hispanics who used cigarettes after x years from $1985$ is $0.79x+y-40$ Rewrite this function: \begin{align} & 0.79x+y-40=0 \\ & y=40-0.79x \end{align} (II) Step 3: After x years from $1985$, the use of cigarettes would be the same. Therefore, equate both functions and find the value of x. \begin{align} & 38-0.54x=40-0.79x \\ & 0.79x-0.54x=40-38 \\ & 0.25x=2 \\ & x=8 \end{align} Since, $\left( x=8 \right)$ then after $8$ (1993) years the use of cigarettes was the same for Hispanics and African Americans. Step 4: Compute the percentage of the use of cigarettes for African Americans in 1993 by substituting $\left( x=8 \right)$ in equation (I) as shown below: \begin{align} & y=38-0.54\times 8 \\ & =38-4.32 \\ & =33.68 \end{align} Since, $y=33.68$. Therefore, the percentage of cigarette use by African Americans in the year $1993$ is $\text{33}\text{.68 }\!\!%\!\!\text{ }$. Step 5: Find the percentage of use of cigarettes for Hispanics, also in $1993$, by substituting $\left( x=8 \right)$ in equation (II) as shown below: \begin{align} & y=40-0.79\times 8 \\ & =40-6.32 \\ & =33.68 \end{align} Since, $y=33.68$ Therefore, the percentage of use of cigarettes for Hispanics in the year $1993$ is $\text{33}\text{.68 }\!\!%\!\!\text{ }$. Thus, the use of cigarettes for both groups was the same after $8$ years from the year $1985$.